Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{-6a^2 - 18a + 420}{7a^2 + 77a + 70}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {-6(a^2 + 3a - 70)} {7(a^2 + 11a + 10)} $ $ r = -\dfrac{6}{7} \cdot \dfrac{a^2 + 3a - 70}{a^2 + 11a + 10} $ Next factor the numerator and denominator. $ r = - \dfrac{6}{7} \cdot \dfrac{(a + 10)(a - 7)}{(a + 10)(a + 1)}$ Assuming $a \neq -10$ , we can cancel the $a + 10$ $ r = - \dfrac{6}{7} \cdot \dfrac{a - 7}{a + 1}$ Therefore: $ r = \dfrac{ -6(a - 7)}{ 7(a + 1)}$, $a \neq -10$